Problem 22 - A Graphical Approach to College Algebra

Question

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=t-2, y=\frac{1}{2} t^{2}+1, \text { for } t \text { in }(-\infty, \infty)$$

Step-by-Step Solution

Answer

The rectangular equation is \(y = \frac{1}{2}x^2 + 2x + 3\) for \(x \in (-\infty, \infty)\).

Step 1: Express Parameter t in Terms of x

We start with the equation for\(x\):\[x = t - 2\].\[\]To express \(t\) in terms of \(x\), solve for \(t\):\[\]\[t = x + 2\].

Step 2: Substitute Expression for t into Equation for y

Now take the expression for \(t\) found in Step 1, \(t = x + 2\), and substitute it into the equation for \(y\):\[\]\[y = \frac{1}{2} t^2 + 1\].\[\]After substitution, it becomes\[\]\[y = \frac{1}{2} (x + 2)^2 + 1\].

Step 3: Simplify the Equation for y

Expand \((x+2)^2\):\[\]\[(x + 2)^2 = x^2 + 4x + 4\].\[\]Substitute this back into the equation for \(y\):\[\]\[y = \frac{1}{2}(x^2 + 4x + 4) + 1\].\[\]Distribute the \(\frac{1}{2}\):\[\]\[y = \frac{1}{2}x^2 + 2x + 2\].\[\]Add the constant 1 on the outside:\[\]\[y = \frac{1}{2}x^2 + 2x + 3\].

Step 4: Determine the Appropriate Interval for x

The original parameter \(t\) was defined for \((-\infty, \infty)\), which means there are no restrictions on \(t\). The transformation we used to express \(t\) in terms of \(x\), \(t = x + 2\), implies that \(x = t - 2\) and since \(t\) covers all real numbers, \(x\) also covers all real numbers. Thus, the interval for \(x\) is \((-\infty, \infty)\).

Key Concepts

Parametric Equations

Parametric equations are a fascinating mathematical concept that describe a curve by using a parameter—typically denoted as \( t \). Instead of directly relating \( x \) and \( y \) like in a standard rectangular equation \( y = f(x) \), parametric equations express both variables in terms of \( t \). For instance, in the exercise where \( x = t - 2 \) and \( y = \frac{1}{2}t^2 + 1 \), \( t \) is our parameter. This method allows for more flexibility and can describe more complex curves, even when \( t \) may not be a simple function.

Parametric equations are useful in various applications, such as kinematics, where they can represent an object's position over time. The process of transitioning from parametric to rectangular involves eliminating the parameter \( t \), which requires expressional substitutions and algebraic techniques. This manipulation enables us to obtain a single equation in terms of \( x \) and \( y \), like we did in this problem where the final rectangular equation is \( y = \frac{1}{2}x^2 + 2x + 3 \).

Interval Notation

Interval notation is an efficient way of expressing a range of values that a certain variable can take. This notation is particularly relevant when discussing the domain of a function or interval for a curve in parametric form.

For this exercise, the parameter \( t \) was defined over the interval \((-\infty, \infty)\), meaning it spanned all real numbers with no limitations. When we changed this to our \( x \) variable, the transformation \( x = t - 2 \) logically retains this span since \( t \) can be any real number. So, the interval for \( x \) is also \((-\infty, \infty)\).

In interval notation, parentheses \( ( ) \) indicate that an endpoint is not included or the range is open-ended, while brackets \( [ ] \) would signal inclusion or closure of a boundary. Understanding interval notation is crucial in determining the applicability of mathematical solutions across different domains and provides clarity when communicating ranges.

Algebraic Manipulation

Algebraic manipulation is at the heart of transitioning parametric equations into rectangular form. It involves solving equations for a variable, substituting expressions, and simplifying to arrive at an understandable form.

In the given problem, we first expressed \( t \) in terms of \( x \) with the equation \( t = x + 2 \), which involved a simple algebraic rearrangement. This expression was then substituted into the parametric equation for \( y \): \( y = \frac{1}{2}t^2 + 1 \).

Next, expanding the squared term \( (x + 2)^2 = x^2 + 4x + 4 \) required applying the distributive property. Substituting into the equation for \( y \) and simplifying led to the final rectangular equation \( y = \frac{1}{2}x^2 + 2x + 3 \).

Steps like these highlight the skill of algebraic manipulation, ultimately allowing us to derive a clear relationship between \( x \) and \( y \) without involving the parameter \( t \). Mastery of these techniques ensures a robust understanding of mathematics beyond initial expressions, enabling one to reformat and resolve equations as needed for further analyses.